Survival analysis of a stochastic impulsive single-species population model with migration driven by environmental toxicant

Considering the influence of environmental toxicant on population migration between patches, we propose and study a stochastic impulsive single-species population model with migration driven by environmental toxicant in this paper. We first discuss the existence and uniqueness of global positive solutions of the model by constructing the Lyapunov function. Then, we obtain sufficient conditions for extinction, stochastic persistence and persistence in the mean of the single-species population. Finally, we present some numerical simulations to illustrate our results. These results provide insights for the conservation and management of species in polluted environments.

where r > 0 and a > 0 stand for the population growth rate and the intra-specific competition coefficient of population. D > 0 is the diffusion coefficient. H and h are sizes of the non-nature reserve and the nature reserve. And then, the extinction and permanence in the mean of single species under fluctuated environments were also studied by Zou et al. 12,13 and Dieu et al. 14 . Based on the model in 12 , Wei and Wang 15 established the following stochastic single-species model with migrations between two patches.
where d 12 ≥ 0 stands for the migration rate of the population from the non-nature reserve (patch 1) to the nature reserve (patch 2), d 21 ≥ 0 stands for the migration rate of the population from the nature reserve to the nonnature reserve. E i denotes the hunting rate in the i-th patch, and E 1 ≫ E 2 . B(t) is standard Brownian motion. In 15 , authors assumed that the number of individuals of a species in the nature reserve is larger than that in the non-nature reserve, and sufficient conditions for the extinction and persistence in the mean of population were obtained. However, it is not difficult to find that the growth rate, the intra-specific competition coefficient and the intensity of white noise in two patches are the same, so the results obtained in 15 are not suitable for the general situation. Therefore, we need to further discuss the influence of population migration on the survival of single-species.
With the rapid development of human society, a large number of toxic substances and pollutants are discharged into the ecosystem, seriously polluting the ecological environment and threatening the survival of species. Such as heavy metal pollution, and water pollution caused by crop fertilization and pesticide application. Therefore, it is most important to investigate the survival and extinction of species in a polluted environment. In recent years, many excellent results have analyzed the effects of toxicant discharged into the environment from modern industry and modern agriculture on population by establishing models [16][17][18][19][20][21] . But, these models mainly discussed the effect of pollutants on the population growth rate. As we all know, many creatures in nature have good sensory organs and highly differentiated nervous systems, and they can respond to information in the environment accordingly. For example, in agricultural production, many pests will choose to escape from the pesticide-treated environment due to the stimulation of chemical pesticides, and then seek a new environment conducive to population growth, this may be one of the reasons for inducing the resurgence of pest populations and the emergence of pest resistance. Therefore, it is necessary to consider the effect of environmental toxicant on population migration. Wei el at. 20,21 proposed two single-species population models with physiological effect, where the "physiological" effect is described as self-protection by organisms in highly polluted environments to reduce the effective contact between the organism and the polluted environment. However, few studies have considered the influence of environmental toxicant on population migration between patches. In this paper, we assume that toxins are emitted in regular pulses, a common example being the use of pesticides, and propose a deterministic single-species population model with migration driven by environmental toxicant as follows: Here x i (t) denotes the density of population in patch i. c e (t) and c o (t) represent the concentration of toxicant in the environment and organism at time t respectively. f > 0 represents the uptake rate of toxicant from the environment by the population in patch 1. (g + m)c o (t) describes loss due to egestion and metabolic process at time t. b ≥ 0 and γ > 0 represent the pulse input amount of toxins and the pulse input period of toxicant respectively. hc(t) represents the total lose at time t from the system environment including processes such as biological transformation, microbial degradation, volatilization and photosynthetic degradation. δc o (t) represents the lethal rate of toxins in the organism to the population in patch 1. In this paper, we adopt a Holling-III response function ρd 12 c 2 e 1+αc 2 e to describe the influence of toxicant concentration in patch 1 on population migration. ρd 12 is described as the migration rate of the population in patch 1 to patch 2 due to the stimulation of toxicant in patch 1, and α > 0 denotes the sensitivity of population to environmental toxicant.
On the other hand, the population is inevitably affected by various factors in the environment, for example, changes in temperature, climate and weather. May 22 showed that the birth rates, carrying capacity, and other parameters involved in the system can be affected by environmental noise. In order to better understand the dynamic behaviors of the population models, many researchers introduced random perturbations into deterministic models to show richer and more complex dynamic properties [24][25][26][27][28][29][30][31] . Motivated by the above studies, we suppose that environmental noises mainly affect the growth rate r ie of system (1) in this paper, according to the central limits theorem, we usually use an average value plus an error term satisfying the standard normal distribution to estimate a value 25,26 , that is, where r ie is a positive constant, dB i (t) dt is the a Gaussian white noise, B i (t) represents the standard Brownian motion defined on the complete probability space (�, F , {F t } t≥0 , P) with {F t } t≥0 satisfying the usual conditions 23 . σ i is the intensity of the white noise. There is another possible form of modeling for r ie in a randomly-varying environment, we introduce the Ornstein-Uhlenbeck process (also called as mean-reverting process) [21][22][23][24][25][26][27] , and it has the following form where r ie , ξ i and µ i are positive constants, µ i is the speed of reversion and ξ i is the intensity of the white noise. Solving the stochastic Eq. (2), from studies 21-27 , we have Modifying the deterministic model (1), we propose the following stochastic impulsive single-species population model with migration driven by environmental toxicant Because the solutions of c 0 (t) and c e (t) can be solved by the third and fourth equations of (3), we only consider the following system Remark 1 Because each of c o (t) and c e (t) is a concentration, c o (t) and c e (t) must satisfy the inequalities 0 ≤ c o (t) ≤ 1 and 0 ≤ c e (t) ≤ 1 for t ≥ 0 . Therefore, throughout this article, we assume that f ≤ g + m and b ≤ 1 − e −hγ . and

Preliminaries
Lemma 1 (see 31 ) Consider the following model corresponding to model (3)

Main results
In order to study the long-time behaviors of the model (4), we first discuss the existence and uniqueness of global positive solutions to the stochastic differential equation (SDE) (4).

Existence and uniqueness of the positive solution for SDE (4).
Theorem 1 For any given initial value (4), and the solution x(t) will remain R 2 + with probability 1.
Proof Because the coefficients of the SDE (4) are locally Lipschitz continuous, there must be a unique local solution x(t) in [0, τ e ) for any given initial value x(0) ∈ R 2 + , where τ e denotes the explosion time. Therefore, we need to prove τ e = +∞ a.s. in the following. Let N 0 be large enough such that x(0) remains in the interval [ 1 N 0 , N 0 ] . For every N ≥ N 0 , define the stopping time Clearly, τ N is increasing as N → +∞ . Letting τ ∞ = lim N→∞ τ N , thus, τ ∞ ≤ τ e a.s. In the following, we only need to prove τ ∞ = +∞, a.s. We next employ the reduction to absurdity to prove it. If the conclusion is not true, then there are T > 0 and ǫ ∈ (0, 1) such that P{τ ∞ < T} > ǫ . Accordingly, there is a positive integer N 1 ≥ N 0 such that for any N ≥ N 1 , P{τ N ≤ T} ≥ ǫ . Define a C 2 -function V : R 2 + → R + as follows: Using Itô ′ s formula, we have , , , and then taking expectation obtain that For any ω ∈ � N , we get that at least one of x 1 (τ N , ω) and From (7), we have Letting N → +∞ , leads to the contradiction: Therefore, we obtain τ ∞ = +∞ , a.s.

Stochastic permanence.
Lemma 3 For any given initial value x(0) ∈ R 2 + , there must be a K(p) > 0 such that the solution x(t) of SDE (4) satisfies Let y(t) = EV (x(t)) , from (8), we have p (x(s))ds, . This ends the proof.

Remark 2 From Lemma 3, we know that there exists a
Applying Itô 's formula, we have I f min{r 1e − δc M , r 2e } > 0.5σ 2 , w e c a n t a k e a n ǫ > 0 s m a l l e n o u g h s u c h t h a t ř = min{(r 1 (t)) * , (r 2 (t)) * } = min{r 1e − δc M , r 2e } > 0.5σ 2 + ǫ . Moreover, we can also select a θ > 0 such that It follows from (12) that .
We will prove in the following that for any ǫ > 0 , there is a H 2 (ǫ) > 0 such that lim inf t→+∞ P{(x 1 (t) + x 2 (t)) ≤ H 2 } ≥ 1 − ǫ. According to Lemma 3 and the Chebyshev's inequality, this result can be easily confirmed.
This completes the proof. (4) with initial value x(0) ∈ R 2 + . If any of the following conditions is true,

Theorem 3 Let x(t) be a solution of SDE
Then the single-species population goes to die out, that is, lim t→+∞ x i (t) = 0, a.s.
Case(ii) : If r * 1 < r 2e , we take a θ 1 > 1 , from (19), we have Let (θ 1 , 1 ) be the solution of the following equations and which implies that r * it is easy to calculate that the quadratic equation (26) has two real roots: And because f (−q) = −d 21 q > 0 , it is easy to see that 0 < p 1 < −q , further, From (24), we obtain that we choose an ǫ small enough such that 1 < 0.5σ 2 , from (27), we also conclude that lim t→+∞ x i (t) = 0, a.s., i = 1, 2.

Remark 3
From the proof of Theorem 3'(ii), we know that species goes to extinction when r * 1 < 0 and d * 12 r 2e + r * 1 d 21 − r * 1 r 2e < 0 , which is independent of the intensity of the noise.

Remark 4
If ρ = 0 , that is, without considering the influence of environmental toxicant concentration on population migration. From Theorem 3, we obtain that single-species population will be extinct if Permanence in the mean. In this subsection, we aim to analyze the permanence in the mean of SDE (4).

Conclusions
With the rapid growth of economy, a large number of toxic substances are discharged into the ecosystem, which seriously threatens the survival of species and human beings. Based on its theoretical and practical significance, stochastic population models with impulsive toxicant input and stochastic single-species population models with migration have attracted many scholars' attention (see, e.g., [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31] ). Up to our knowledge, few studies have considered the influence of environmental toxins on population migration between patches. In this paper, we propose and study a stochastic single-species population system with migration driven by environmental toxicant and impulsive toxicant input. We prove the existence and uniqueness of the global positive solution of SDE (3) by constructing the Lyapunov function, and analyze the boundedness of the p-moments of the solution. And then, we obtain sufficient conditions for population extinction, stochastic permanence and permanence in the mean. There results show that the intensity of white noise ξ i , the speed of reversion µ i , the pulse input period of toxicant γ , the toxicant input amount each time b and the population migration between patches play a very important role on the survival of the population, see Figs. 1 and 2. Finally, we also study the stochastic singlespecies population model with migration between two non-polluted patches, and give the sufficient conditions for population extinction and permanence.
On the other hand, there are many interesting problems that deserve further study, for example, the existence and uniqueness of the ergodic stationary probability density for system (3) (see 34,35 ), and many more realistic but complex models should be formulated (see 36 ). In addition, the telegraph noise can be illustrated as a switching between two or more regimes of environment, which differ by factors such as nutrition or as rain falls 37,38 , which is memoryless and the waiting time for the next switch has an exponential distribution, we can use a finite-state Markov chain to simulate regime switching in here. Therefore, it is interesting to introduce the telegraph noise into model (3). We shall also consider this question in our future work.